Non-equilibrium quantum physics has become one of the central fields of
research in particular due to an impressive degree of control in recent cold
atom and quantum optics experiments. On the theoretical side, the topic
provides a very fruitful playground where many seemingly diverse fields of
theoretical and mathematical physics meet and overlap, such as condensed matter
theory, statistical mechanics and field theory, string theory, quantum
information, and dynamical systems. Having relevant and nontrivial exactly
solvable models displaying key universal phenomena is a crucial aspect of every
successful theory. In non-equilibrium quantum interacting many-body systems,
such exact solutions have been very rare, and only very recently substantial
progress has been made. In this talk I will describe one thread of recent
progress, which started from exact solutions of boundary driven master
equations of interacting quantum spin chains and ended up in discovering new
families of quasilocal conservation laws. The latter are relevant for
describing equilibration after quantum quenches and rigorously establishing
ballistic or diffusive high temperature transport in integrable systems.